Should I be suspicious?

[Robert2016]

Are there no women that take their kids to the park?

Why didn't she invite you to the community event?

What was the community event that she invited him to attend with her?  If he accepts her invitation, I think you should tag along (out of sight) to observe them together and to see where they go afterward.

Does the OM work during the day?  How is he available for play dates?

Big red flag. Especially after 3 weeks of no play dates or texts.   Her recent text is evidence that the OM is on her mind and it's a relationship she wants.   And inviting him to meet her is something she would not do if he came onto her - and she rejected him.

Sounds like he was out of town or otherwise busy.

When you joked about him being her boyfriend, a more appropriate response (after first asking for your permission) would be to ask if you're STILL ok with her meeting with him (but she chose not to open that door). 

At this moment in time, she likely means it when she denies any type of relationship, emotional or physical.   However, why him vs another mom; and she wouldn't reach out to him if she didn't enjoy his attention/company.  She likely enjoys his attention and feels 100% in control (especially with the girls present) - and that it will not escalate to an affair.  Unfortunately, she's playing with fire.

This OM is high risk and not someone your wife should be reaching out to regularly (for play dates or to attend a community event with). 

Lot's of connections start innocent with:  she says he's going through a divorce and needs someone to talk to.  In reality she's neither qualified nor is it her business.

Dr Shirley Glass (among  other authors) has a book about 'boundaries' and behavior that protect a marriage from 'friends'.  I suggest you both read up on boundaries and then in a non confrontational way discuss how boundaries can protect your marriage.

Put a gps in her car and a VAR.  Then IMO you should tell your wife that you changed your mind about her spending time with the OM.  If she refuses, then you have your answer.  Plus you can track her movements and calls to the OM.